IMAGE  EVALUATION 
TEST  TARGET  (MT-3) 


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Photographic 

Sciences 
Corporation 


23  WEST  MAIN  STREET 

WEBSTER,  N.Y.  14580 

(716)  872-4503 


M^^m 


•^-■j 


CIHM/ICMH 

Microfiche 

Series. 


CIHIVI/ICMH 
Collection  de 
microfiches. 


Canad    1  Institute  for  Historical  Microreproductions  /Institut  Canadian  de  microreproductions  historiques 


;i 


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Technical  and  Bibliographic  Notes/rtJotes  techniques  et  bibliographiques 


The  Institute  has  attempted  to  obtain  the  best 
original  copy  available  for  filming.  Features  of  this 
copy  which  may  be  bibliographically  unique, 
which  may  alter  any  of  the  images  in  the 
reproduction,  or  which  may  significantly  change 
the  usual  method  of  filming,  are  checked  below. 


n7|    Coloured  covers/ 
L-J    Couverture  de  coulaur 


r~~|    Covers  damaged/ 


Couverture  endommagde 

Covers  restored  and/or  laminated/ 
Couverture  restaurde  et/ou  pellicul^e 

Cover  title  missing/ 

Le  titre  de  couverture  manque 

Coloured  maps/ 

Cartes  g^ographiques  en  coulaur 

Coloured  ink  (i.e.  other  than  blue  or  black)/ 
Encre  de  couleur  (i.e.  autre  que  bleua  ou  noire) 

Coloured  plates  and/or  illustrations/ 
Planches  et/ou  illustrations  en  couleur 

Bound  with  other  material/ 
Relii  avec  d'autres  documents 


D 


D 


D 


Tight  binding  may  cause  shadows  or  distortion 
along  interior  margin/ 

La  re  liure  serr^e  peut  causer  de  I'ombre  ou  de  la 
distorsion  le  long  de  la  marge  intdrieure 

Blank  leaves  added  during  restoration  may 
appear  within  the  text.  Whenever  possible,  these 
have  been  omitted  from  filming/ 
II  se  peut  que  certainss  pages  blanches  ajouties 
lors  d'une  restauration  apparaissent  dans  le  texte. 
mais,  lorsque  cela  dtait  possible,  ces  pages  n'ont 
pas  M  filmies. 

Additional  comments:/ 
Commentaires  suppldmentaires; 


L'Institut  a  microfilm^  le  meilleur  exemplaire 
qu'il  lui  a  6ti  possible  de  se  procurer.  Les  details 
de  cet  exemplaire  qui  sont  peut-dtre  uniques  du 
point  de  vue  bibliographique,  qui  peuvent  modifier 
une  image  reprodulte,  ou  qui  peuvent  exiger  une 
modification  dans  la  mdthode  normale  de  filmage 
sont  indiquis  ci-dessous. 

□    Coloured  pages/ 
Pages  de  couleur 

□    Pages  damaged/ 
Pages  endommagies 

□    Pages  restored  and/or  laminated/ 
Pages  restauries  et/ou  pelliculdes 

0'Pagea  discoloured,  stained  or  foxed/ 
Pages  ddcolor^es,  tachetdes  ou  piqudes 

□    Pages  detached/ 
Pages  ddtachees 

r^l^howthrough/ 
L-J    Transparence 

r~~]    Quality  of  print  varies/ 


Quality  indgale  de  ('impression 

Includes  supplementary  material/ 
Comprend  du  materiel  supplementaire 

Only  edition  available/ 
Seule  Edition  disponible 


D 


Pages  wholly  or  partially  obscured  by  errata 
slips,  tissues,  etc.,  have  been  refilmed  to 
ensure  the  best  possible  image/ 
Les  pages  totalement  ou  partiellement 
obscurcies  par  un  feuillet  d'errata,  une  peiure, 
etc.,  ont  M  filmies  d  nouveau  de  facon  d 
obtenir  la  meilleure  image  possible. 


The 
to  th 


The 
poss 
of  th 
filmi 


Orig 
begi 
the  I 
sion 
othe 
first 
sicn, 
or  ill 


The 
shall 
TINl 
whic 

Map 
diffe 
entir 
begii 
right 
requ 
mett 


This  item  is  filmed  at  the  reduction  ratio  'hacked  below/ 
Ce  document  est  filmi  au  taux  de  reduction  indiqui  ci-dessous. 
10X  14X  18X  22X 


UA 


lOA 


y 


20X 


26X 


30X 


24X 


28X 


32X 


The  copy  filmed  here  has  been  reproduced  thanks 
to  the  generosity  of: 

Metropolitan  Toronto  Library 
Canadi&n  History  Department 


The  images  appearing  here  are  the  best  quality 
possible  considering  the  condition  and  legibility 
of  the  original  copy  and  in  keeping  with  the 
filming  contract  specifications. 


Original  copies  in  printed  paper  covers  are  filmed 
beginning  with  the  front  cover  and  ending  on 
the  last  page  with  a  printed  or  illustrated  imprss- 
sion,  or  the  back  cover  when  appropriate.  All 
other  original  copies  are  filmed  beginning  on  the 
first  page  with  a  printed  or  illustrated  Impres- 
sion, and  ending  on  the  last  page  with  a  printed 
or  illustrated  impression. 


The  last  recorded  frame  on  each  microfiche 
shall  contain  the  symbol  ^»- (meaning  "CON- 
TINUED"), or  the  symbol  V  (meaning  "END"), 
whichever  applies. 

Maps,  plates,  charts,  etc.,  may  be  filmed  at 
different  reduction  ratios.  Those  too  large  to  be 
entirely  included  in  one  exposure  are  filmed 
beginning  in  the  upper  left  hand  corner,  left  to 
right  and  top  to  bottom,  as  many  frames  as 
required.  The  following  diagrams  illustrate  the 
method: 


1 

2 

3 

L'exemplaire  fiimd  fut  reproduit  grSce  d  la 
gun6rosit6  de: 

■Metropolitan  Toronto  Library 
Canadian  History  Department 


Les  images  suivantes  ont  dtd  reproduites  avec  le 
plus  grand  soin,  compt«t  tenu  dc  la  condition  et 
de  la  netteti  de  l'exemplaire  filmi,  et  en 
conformity  avec  les  conditions  du  contrat  de 
filmage. 

Les  exemplaires  originaux  dont  la  couverture  en 
papier  est  imprimie  sont  film6s  en  commengant 
par  le  premier  plat  et  en  terminant  soit  par  la 
dernidre  page  qui  comporte  une  empreinte 
d'impression  ou  d'illustration,  soit  par  le  second 
plat,  salon  le  cas.  Tous  les  autres  exemplaires 
originaux  sont  filmds  en  commen9ant  par  la 
premidre  page  qui  comporte  une  empreinte 
d'impres&ion  ou  d'illustration  et  en  terminant  par 
la  dernidre  page  qui  comporte  une  telle 
empreinte. 

Un  des  symboles  suivants  apparaftra  .rjr  la 
dernidre  image  de  cheque  microfiche,  selon  le 
cas:  lo  symbole  — ►  signifie  "A  SUIVRE",  lf> 
symbole  V  signifie  "FIN". 

Les  cartes,  planches,  tableaux,  etc.,  peuvent  dtre 
filmis  d  des  taux  de  reduction  diff^ronts. 
Lorsque  le  document  est  trop  grand  pour  dtre 
reproduit  en  un  seul  clich6,  il  est  film6  d  partir 
de  Tangle  sup6rieur  gauche,  de  gauche  d  droite, 
et  de  haut  en  bas,  en  prenant  le  nombre 
d'images  ndcessaire.  Les  diagrammes  suivants 
illustrsnt  la  mdthode. 


1 

2 

3 

4 

5 

6 

liiwwwilli*.  .•  *,...i||ILl"U!l.,i!,Jl!PIUIl|"l< 


A    LETTER 

TO    THE 

PRESIDENT    OF   THE   U.    STATES    OF   AMERICA 

"IIOI'OSING    A   METHOD   WIIERKBY  THE    JIEUITS    OF 
THE  CONFLICTINS  CLAIMS  OF 

THE   UNITED  STATES   AND   GREAT  BRITAIN, 

ON   THE   DISPUTED  FRONTIER, 

MAir     UE      CORRECTLY      EXAMINTCD      ANT)      DRT  KR  M I N  En,     HV      SCIE\TIKIC 

ruiwuiPL.i:s,    coMuivT-.n    with    indisputable    data, 
grol      a:ii    in    equity; 

togetiif.n  with  an  examination  ok 

THE    TREATY    OF     1783: 

FROM    WHICH    IS    DERIVED    A    DI'.MOXSTRATION    OF    THE    FALSEHOOD    OF 

THE    BRITISH    CLAIM. 
By   JOHN   LEE, 

OECYPHERER   OF  THE    CHARTER   OF    OMAOH,  AND    AUTHOR   OF   PROPORTIONAL 

FORMUL/E,  THEORV    OF    LOMUT  f  J)  K ,    AND    A    TKKATISE    ON    THE 

ERRORS    OF    Tin:    RECTANGULAR    SURVEY. 


"  ^X?.(aS^  if  ari],'^eoni  li^^i  tl>ov  n'ni'ona  ttvg 
Eadhty,  Tou  di  rt   nn'/J.oi  mav{ilnxorT^  ur^Qiunoi, 
Ka'i  TE  ni/keig  »(Tuo)(T£."  —  Iliad,  Lil).  XIll. 

"  r,  wisdom,  dwell  with  prudence,  and  find  out  knowledge  of  witty  inven- 
tions."—  Proverbs.,  viii.  12. 

"  Wisdom  is  better  than  weapons  of  war."  —  Ecclesiastes,  ix.  18. 


CAMBRIDGE: 

METCALF,    TORRY,    AND    BALLOU. 

1839. 


3** 


.ymmam 


7 


3*^ 


->, 


*   * 


.  -• 


«  '.*  > 


!A<-D  E  T  T.E  Jl.   :  .^ 


TO    THE 


r..A: 


PRESIDENT    OF  ^'IIE    U.    STATES  ^OF    AMERICA; 

PROPOSING    A   MKTIIOO    WIII'.UEBY   THE    MERITS    OF 

. » ..■"»•  - .    TiHB  efrNi'i.ieTi?*(i  «la ms<toF 


THE    UNITED   STATES   AND   GREAT   BRITAIN, 
ON   THE   disputed' FROxXTIEIl, 


•..* 


MAY     BE      COnRKCTLY      FXAMINKD      A.VD      DETEUMINED,     BY      SCIENTIKIC 

PRINCII'LES,      COMUIXED      WITH      I  M)  I  S  [' UT  ABLE      DATA, 

GROUNDED     IX      EQUITV; 

TOGETHER    WITH    AN    EXAUIXATION    OF 

THE    TREATY    OF     178  3: 

FROM     WHICH    IS    DERIVED     A    DEMONSTRATION    OF    THE    FALSEHOOD    OF 

THE    BRITISH    CLAIM. 


By    JOHN    LEE, 

DECYPHERER    OF   THE    CHARTER    OF    OMAGH,  AND    AUTHOR    OF    PROPORTIONAL 

FORMULAE,   THEORY    OF    LONGITIDE,    AND    A    TREATISE    ON    THE 

ERRORS    OF    THE    RECTANGULAR    SCRVEY. 


End>.'„y,  70V   (5*   rs    rn/O.ol   e:cavoia>tui  r'  urdnmJToi, 
Kal  re    id/fis  iauwat."  —  Iliad,  Lib.  XI 11. 

"  I,  wisdom,  dwell  with  prudence,  and  find  out  knowledge  of  witty  inven- 
tions."—  Proverbs,  viii.  12. 

"  Wisdom  is  better  than  weapons  of  war."  —  Ecclesiastes    ix.  18, 


CAMBRIDGE: 

METCALF,   TORRY,    AND    BALLOU. 

1839. 


y 


^^i_  /^ 


^- 


i^L^  ^d 


Entered,  according  to  act  of  Congress,  in  tlio  year  1830,  by  Metcai.f, 
ToRRV,  AND  Ballou,  in  the  Clerk's  OfHco  of  the  District  Court  of  tlic 
District  of  Massachusetts. 


^ 


A    LETTER 


TO    MIS    KXCELLENCV     THE 


PRESIDENT    OF  THE    UNITED    STATES  OF  AMERICA. 


> 


teiR  : 

The  conspicuous  majesty  of  Science  is  now  recognised 
from  nation  to  nation,  and  from  clime  to  clime,  tlirongii  the 
circuit  of  the  habitable  universe.  From  her  intellectual 
throne  she  sways  the  destinies  of  men.  In  the  remote  and 
tranquil  scenes  of  rural 'cultivation ;  in  the  busy  throng  of 
traffic  and  the  ceaseless  reveiberating  din  which  pervades  the 
conuTierrial  cities  ;  on  the  vast  and  solitary  surface  of  the 
undulating  deep ;  in  the  barriers  of  eternal  polar  ice,  or  the 
impenetrable  silent  horrors  of  subterrestrial  gloom  ;  through 
all  the  confines  of  the  sublunary  world,  she  clauns  ubiquity 
of  empire.  That  portion  cf  our  being,  which  we  inherit  from 
divinity,  has  been,  by  the  discipline  of  Science,  developed  in 
such  magnificence  of  energy  as  clearly  marks  its  origin  ; 
while  the  petty  powers  of  our  animal  constitution  have  sunk 
to  comparative  nonentity.  In  the  conlhct  of  battle,  and  the 
agonizing  shock  of  war,  the  achievements  of  corporeal  prow- 
ess are  beheld  no  more  ;  the  mail-protected  rank  of  heroes  has 
dwindled  into  puny  insignificance,  before  the  vast  machinery 
of  destruction  which  Science  has  created  and  arrayed  upon 
the  martial  field ;  she  impels,  with  invisible  and  superhuman 
arm,  the  flame-winged  missiles  which  annihilate  legions,  and 
crush  the  pride  of  castellated  bulwarks  with  irresistible  and 
ruinous  prostration. 

This  development  of  the  mental  energy  of  man,  though  its 
recent  progress  to  maturity  be  characterized  with  gigantic  and 


marvellous  rapidity,  is  no  preci|iitato  result  of  late  or  sndden 
causes.  It  is  tli(!  gradual  yrowtli  and  liiial  iVuctificutiou  of 
the  trtc  of  KiKiiv/ri/i'f,  tliroiiijii  tlic  total  extent  of  a  period 
wliicli  lias  nearly  eoiupreheiidcil  six  thousand  years.  Through 
iho  reckless  and  harharian  atheism  of  lawless  unconfedcraied 
man  ;  through  the  mystic  incantations  and  fantastic  horrors 
which  mark  the  fearful  rites  of  idolatrous  bewildered  nations; 
and  in  later  times,  ilirough  all  the  seductive  sophistry  and 
blasphemous  imj)i(.'ty  of  audacious  and  jicrverse  philosophy  ; 
hke  the  vegetative  bud  which  survives  the  changes  of  the 
atmosphere,  the  intellectual  germ  has  resisted  every  ungenial 
inlluence  of  place  and  time,  ami  preserved  its  undecayed  vi- 
tality, to  attain  the  plenitude  and  lustre  of  autumnal  exuber- 
ance and  bloom. 

And,  tlierefore,  when  contrasted  with  the  flruning  splendor 
■which  illuminates  the  annals  of  science  in  the  present  age, 
the  page  o[  ancient  history  is  not  coiiiplctcli/  dcso\iitr.  and  dark. 
Through  the  mist  of  intervening  ages,  the  achievements  of 
the  peerless  Archiinrdcs  ]n-esent  a  monumental  prototype  of 
almost  every  marvellous  ])lienomenori,  which,  by  the  aid  of 
science,  is  created  on  the  theatre  of  nations  in  the  jiresent  day. 
His  caustic  apparatus  and  projectile  engines  refer  the  imagiu- 
ution,  by  a  simultaneous  and  triple  analogy,  to  the  sudden 
flash  and  heavy  roar  of  the  cannon,  to  the  lightning  and 
thunder  of  the  electii'^.  machine,  and  the  irresistible  energy 
which  results  from  the  concentration  of  the  optic  ray.  His 
investigations  in  tlie  theory  of  motion,  e(|uilibrium,  and  im- 
pulse, have  served  as  a  basis  for  all  that  we  now  know  in 
mechanical  science,  through  all  its  modifications  and  depart- 
ments, comprehending  tlie  multitudinous  motive  powers  and 
impulsive  agencies  of  solid  and  fluid  materials.  And  in  bright 
and  glorious  addition  to  all  the  preceding  discoveries,  his  dar- 
ing and  sublime  excursions,  in  the  intellectual  region  of  pure 
Geometry^  through  unexplored  and  solitary  tracts  of  knowl- 
edm;,  but  copious  and  luxuriant  in  the  intrinsic  and  immortal 
beauty  of  the  immntahle  valnrc  of  tilings ;  these  intrepid 
flights  of  reason  have  directed  the  research  of  every  subse- 
quent adventurer  on  that  boundless  and  eternal  field. 


5 


Sir,  it  is  not  nocessury  hero,  to  (Minmorato,  in  detail,  tlie  im- 
portant sorvioes  for  wliich  cclehrily  is  duo  to  othvr  niunerous 
and  iliiisirioMs  names,  which  adorn  the  rccortUxl  catalogue  of 
scientific  labors  and  lore,  through  the  long  and  shadowy  tract 
of  time  comprehended  in  ancient  liistory  ;  though  the  mental 
facidties  of  yoin-  Excellency  must  have  been,  for  years,  indis- 
pensably j)re-occupi(!d  with  all  the  responsibilities  and  cares 
attached  to  the  superintendence  of  the  political  allairs  of  a  great 
and  rising  nation;  I  consider  it  yet  a  presumable circimistance, 
that  all  the  classic  recollections,  which  nifer  to  the  develop- 
ment and  progress  of  the  junnan  mind  from  early  imbecility 
to  intellectual  manhood,  have  not  completely  faded  from  your 
memory. 

Though  tlie  extraordinary  and  startling  ]>roblem,  the  per- 
formance of  Avhich  is  the  professed  and  immediate  object  of 
the  present  letter,  be  one  which  is  indeed  unexampled  in  the 
history  of  -cience,  the  preceding  rapid  sketch  of  the  continu- 
ous movements  and  incessant  researches  of  the  lunnan  mind, 
will,  I  am  disposed  to  believe,  diminish  the  deep  surprise, 
with  which,  for  the  first  time,  your  Excellency,  and  the  public 
would  otherwise  receive  the  intimationof  the  attempted  solution 
of  Ibis  prol)lem.  l>y  ^  sim[ile  and  direct  pursuit  of  the  ideal 
train  of  associations  presented  in  the  preceding  sketch,  the 
solution  of  this  problem  will  easily  appear  to  ho  part  and  par- 
cel of  the  connected  results  of  the  uiupiiet  and  aspiring  ope- 
rations of  tb,a  'general  Iniwnn  tn'unl  :  of  that  primeval, 
celestial,  and  elastic  impidse,  whereby  our  universal  race  and 
lineage  are  exalted  aliove  the  humility  of  their  native  dust,  to 
tread,  in  intellectual  pre-eminence,  the  ulterior  confines  of  the 
vaulted  universe,  and  contribute  recognition  and  homage  to 
the  invisible  Architect  who  dwells  beyond. 

Sir,  I  am  a  native  and  vohuitary  subject  of  the  British 
sovereign  ;  but  lew  and  doubtful  arc  the  claims  of  gratitude, 
however,  which  the  British  nation,  or  any  other,  has  on  7ny 
iiidkidual  oifeciiohs;  this  country  I  have  sought,  not  as  a 
refuge  for  political  disaffection,  or  a  laboratory  for  the  experi- 
mental career  of  a  political  empiric;  not  as  the  ultimate  re- 
source of  one  who  escapes  from  the  horrors  of  penury,  nor 


t 


yet  as  a  protecting  asylum  for  an  infamous  and  guilty  fugitive; 
I  iiavo  directed  my  steps  uptMi  the  transatlantic  shore,  to 
obtain  a  refuge,  in  rotircniout,  from  scenes  of  adversity,  and 
a  cessation  of  the  miseries  of  mental  distress  ;  to  seek  a  rest- 
ing place  where  sources  of  private  unhapj)incss  may  feel  the 
bulm  of  solitude  ;  where  the  eyes  of  the  depraved  rablilo  may 
never  enjoy  that  luxury  which  is  deemed  intensely  gratefid  to 
maligiiity  and  envy,  when  permitted  to  espy  the  faded  lustre 
of  decayed  resjicctability,  and  the  ruinous  operations  of  disas- 
trous and  reverted  fortune;  where  the  alilieted  proud  man, 
Avhcn  he  feels  the  burning  and  resistless  impulse  to  relieve  and 
vent  his  anguish  by  external  or  internal  tears,  may  weep  in 
secret  and  alone. 

But  while  the  self-dependent  and  secluded  nature  of  my 
present  condition  compels  me  lo  regard  my  feci  It  ij  as  my  otvn, 
and  brook  no  infringement  of  my  personal  rights  from  any 
existing  human  power,  when  the  mcaiui  of  retaliation  are  ac- 
cessible;  I  cling  to  the  belief,  that  I  feel  myself  too  deeply 
bound  by  the  obligations  of  moral  conscience,  by  the  intrinsic 
sanctity  of  justice,  by  tlio  love  of  c(|uity  which  dwells  immu- 
tably inherent  in  the  virtuous  mind,  and  by  the  liuLH-rin"-  at- 
tachments  of  pre-existing  national  sympathy  which  distinguish 
and  adorn  the  liunuui  chaiactcr,  and  interpose  a  true  criterion 
betwixt  the  magnanimous  and  the  servile  spirit ;  I  believe 
myself  too  potently  restricted  by  these  considerations,  to  es- 
pouse the  American  claims  for  the  sake  of  jjopular  favor 
in  considering  the  merits  of  the  great  international  dispute 
which  I  propose  to  examine  in  this  letter. 

I  must  also  observe  to  your  liXcellency,  that,  Avhilc  I  con- 
ceive myself,  in  this  country,  deeply  indebted  to  a  few  indi- 
vidual friends;  yet,  from  the  nation  at  large,  or  any  section, 
great  or  small,  of  the  community,  1  derive  nofavur,  and  there- 
fore adairnvlcdge  no  ohUgation  ;  I  anticipate  no  advantage,  and 
seek  no  benefit,  except  as  the  reward  of  in y  own  exertions; 
and  few  therefore,  I  presume,  can  be  easily  found,  who  are 
better  qualified  by  total  exemption  from  ''fear,  favor,  or  affec- 
tion,'^ than  I  am,  for  the  examination  of  this  great  (piestion. 

Having  offered  the  preceding  remarks  to  your  Excellency, 


• 

for  the  vindication  of  my  own  character,  conduct,  and  motives, 
in  attempting  to  determine  the  merits  and  dispel  the  pcrploxi- 
t'cs  of  this  disjjnte,  I  now  proceed,  wiihont  fuilhcr  preface 
or  delay,  to  enter  on  the  iield  of  c\i)latK'iti()n  and  decisioti. 
And  here,  in  the  andible  and  free  connnnnication,  ana  the 
open  conspicuons  j)rospect  of  the  great  and  general  amphithea- 
tre of  nations,  I  avow  myself  constrained  in  justice,  to  give 
my  direct  and  nneqni vocal  declar  'ion  and  oi)inion  on  the 
side  of  the  American  party  in  this  lif, ration.  1  am  constrained 
to  acknowledge,  that  every  featnro  of  the  Uritisli  claim  so 
visibly  and  palpably  betrays  the  stamp  of  artificial  and  recent 
origin,  as  to  seal  the  'ips  of  every  conscientions  and  enlighten- 
ed man  who  attempts  to  speak  in  its  di;fence,  I  am  constrain- 
ed to  deprei^cuc,  with  horror  and  disdain,  the  dark,  the  danma- 
blc,  perverse,  and  balcfid  doctrine  of  misguided  ethical 
philosophy,  which  tolerates  an  ant  of  iniquity  and  fraud,  ■// 
such  act  be  the  act  of  a  natinn  :  which  designates  extortion  by 
the  title  of  refined  policy*  if  a  nation,  as  one  man,  have  deter- 
mined to  act  in  concert  as  unanimous  ronfidcratc  cvtortioucrs  ; 
which  covers  political  villainy  with  gaudy  epitliets,  and  hides 
the  infamy  and  shame  of  a  people  in  the  frivolous  delusions  of 
insignificant  verbal  vanity  ;  which  encourages  the  national 
and  public  perpetration  of  deeds,  the  doom  and  consequence 
of  Avhich,  to  the  performer,  if  performed  hy  one  individual, 
would  eject  him,  as  a  vile,  abhorred,  and  solitary  fugitive, 
abroad  upon  the  face  of  the  earth.  I  am  constrained  to  ac- 
knowledge, that  if  the  Imi  be  correctly  taken  for  a  symbol  of 
the  nobler  characteristics  of  the  British  nation,  the  same  anal- 
ogy also  too  unhappily  and  closely  prevails,  in  respect  of  some 
other  less  attractive  traits  of  character  which  mark  the  royal 
quadruped  ;  I  am  constrained  to  express  my  fear,  that,  if  con- 
sidered in  reference  to  such  a  comparison,  the  acts  of  that 
nation  have  more  than  once  betrayed  a  propensity  —  royalin 
cupidity,  and  more  than  royal  in  rapacity. 

In  the  examination  of  the  question  of  the  frontier,  which  I 
propose  to  undertake,  on  scientific  principles,  in  the  subsequent 
part  of  this  letter,  many  mathematical  theorems  are  involved, 
of  so  technical  a  nature,  that  I  should  be  guilty  of  gross  and 


* 


.    i 


8 


contemptible 
those  theorems  art 


hypocrisy,   by  pretending    to  presume  that  all 
familiar  to  your  Excellency,  exercised  and 


occupie 


ties 


d  as  you  have  been  for  many  years,  in  the  respoii 
and   dutiuG  of  political  Mlairs. 


isibili- 


For  the   satisfaction    o 
I  pr( 


f 


your  Excellency  concerning  those   technical  inqiuries, 
sume  to  suggest  a  reference,  if  necessary,  to  the  mathematical 
authoriues  of  Yale.  Virginia,  and  West  Point. 

Finally,  Sir,  before  1  undertake  the  proposed  examination, 
I  must  express  the  hope,  that  a  consideration  of  the  nature  of 
my  attempt  will  ten.,  with  additional  force,  to  impress  upon 
your  Excellency,  a  deep  conviction  of  the  incalculable  hcnejits 
resuUino-  to  mankind,  in  every  conccioahle  department  of 
humanl fairs,  from  the  patron  a  ^-e,  difusion,  and  protection  of 
science  '  I  shall  rejoice,  if  enabled  hereafter  to  believe,  that,  by 
any  observation  contained  in  or  suggested  by  this  letter,  the 
attention  of  your  Excellency,  or  that  of  any  other  influential 
man,  has  been  more  closely  directed  to  the  promotion  of  that 

glorious   object. 

But  1  cannot   incur  the    deep   culpability  of  neglecting  to 
inform  your  Excellency,  and  all  my  fellow-men,  that  mathe- 
matical science,  like  every  other  human  pursuit,  has  its  vani- 
ties deceits,  and   snares.     More  especially  in   later  times,  a 
race  of  men  has   appeared  upon  the  field  of  demonstration, 
the  devious  meteoric  aspect  of  whose  intellectual  career  has 
induced  me  to  apply  to  such  indviduals  the  epithet,  Formu- 
larian,  as  a  distinctive  appellation.     These  men  shun  dejim- 
tions,  and  refuse  to  explore  first  principles ,  they  conceal  am- 
biguity by  apparently  precise,  but  latently  equivocal  words; 
they  accomplish,  on  formuUe,  unusual  and  curious  transform- 
ations,   by    unjustly,    though    plausibly,     generah:.ins    the 
ordinary  rules  of  operation,  like  certain  other   philosophers, 
who    having    once    discovered   that   a   circumstance   is  very 
generally  true,   immediately   arm   themselves    with   such   a 
discovery,  as  a  weapon  wherewith  to  deny  the  reahty  oJeve^l> 
case  of  exception  rvhic/i  may  afterwards  occur.     Ihe  absurdi- 
ties of  these  men  are  sometimes  harmless  and  amusing  ;  but 
unhappily,  too  often,  more  calamitous  results  ensue.     Among 
this  visionary  tribe,  La  Place  appears  to   be  pre-eminent,  in 


\ 


of 


propensity  to  mischief,  and  in  power  of  performance.  The 
splendid  fictions  and  marvellous  delusions  of  that  unrivalled 
Formnhirian  are  jiregnant  witii  destruction  ;  they  are  sub- 
versive, in  their  final  tendency,  of  religion,  morality,  and 
social  order;  and  consequently  hostile  to  the  present  and  sub- 
sequent happiness  of  men.  Such  individuals  may  be  rightly 
regarded  as  the  despicable  vermin  which  pollute,  Avhile  they 
prey  upon,  the  intellectual  decayed  remains  of  Archimedes 
and  of  Newton. 

Before  1  undertake  the  proposed  question,  1  shall  finally 
observe  to  your  Excellency,  and  the  public,  that,  concerning 
the  promotion  of  science,  I  am  sorry  to  perceive  the  existence 
of  an  error,  which,  in  politics  and  literature,  is  equally  fatal 
and  disastrous.  This  error  consists  in  the  supposed  nlility  of 
domineerimy  heluivior.  One  of  the  predecessors  of  your 
Excellency,  in  the  exalted  station  which  you  now  occupy, 
unfortunately  tarnished,  by  a  fatal  error  of  that  nature,  the 
lustre  of  his  previous  reputation  ;  but  how  much  more  inde- 
scribably contemptible  is  the  aspect  of  a  dominccrwrr  hidl;/ 
among-  jiiveaih  students,  than  that  of  a  rash  and  arbitrary 
man  directing  the  atlairs  of  a  nation  !  The  only  considera- 
ion,  that  operates  with  resistless  and  perpetual  sway  on  every 
department  and  rank  of  human  society,  is  the  prepossessing- 
native  dignity  which  pccidiarli/  discriminates  the  gentleman; 
and  which  is  completely  incompatible  with  every  vestige  of 
that  repidsive  rudeness,  which  betrays  the  barbarian,  the 
ruilian,  or  the  bear. 

Finally,  I  now  proceed,  according  to  proposal,  to  furnish 
in  detail,  to  your  Excellency,  the  scientific  investigation  and 
solution  of  the  problem  respecting  the  disputed  frontier. 


t 


10 


\  I 


i   \ 


INVESTIGATION     OF    THE     PROBLEM    OF    THE     DISPUTED 

FRONTIER. 


AiiT.  1.     If  two  parlies   agree 


to    discriminate  a  tract  of 


country  into  two  several  shares,  by  a  transverse  boundary,  and 
the  face  of  the  country  present  no  natural  obstacles,  that 
boundary  will,  beyond  all  doubt,  be  UectiUneal ;  for  every 
curvilinear  deflection  or  angular  deviation  would  be  a  source 
of  unprofitable  toil  and  useless  perplexity. 

Art  2.  And  hence  if  ^iuy  part  of  such  boundary  were 
disputed  or  defaced,  it  might  be,  at  any  time,  re-ascertained, 
by  simply  pursidng  the  direction  of  that  part  lohich  is  known. 
Art  3  But  if  natural  obstacles  occur  on  the  face  of  the 
country,  they  will  operate  in  two  ways  ;  first,  by  intercept- 
ing the  course  and  precluding  the  progress  of  the  intended 
rectilineal  boundary  ;  and  secondly,  by  presenting,  m  them- 
selves a  range  of  immutable  and  definite  local  features,  which 
may,  with  facility  and  certainty,  be  taken  as  connecting-points 
through  which  we  may  imagine  a  boundary  to  pass. 

Art    4      Between  the  parties,  however,  a  previous  under- 
standing must  have  existed,   either  strict,  precise,  and  immu- 
table   or   else  restrained  loithin   certain   limits   of  allowable 
adjustment,   concerning    the  proportionate   magnitudes   and 
relative  situations  of  the  two  shares. 

Art  5  And  hence,  in  the  selection  of  such  natural  con- 
nectin-noints,  lo  mark  the  direction  of  a  boundary,  the  pro- 
ceedings will  be  governed  and  alFected  by  this  indispensable 
consideration  ;  so  to  direct  such  a  boundary,  as  to  violate  zn  the 
smallest  possible  degree,  the  understanding  which  existed, 
respecting  the  situations  and  magnitudes  of  the  two  shares. 

Art  6  And  if  the  series  of  natural  points  discontinue,, 
before'this  boundary  has  completed  the  extent  of  its  course, 
and  the  said  boundary,  having  reached  the  last  of  these  points 
emercre  upon  the  uniform  face  of  the  country,  so  that,  beyond 
that  loint,  its  direction  be  optional;  the  only  consideration 
thenceforth  existing,  which  can  regulate  the  course  of  that 
final  part  of  the  boundary,  is  this  :  so  to  direct  this  final  part, 


s. 


n 


m 


as  to  compejimte  for  any  violatinn  of  fhe  aforesaid  previous 
understniidiuL^,  wJiiclt  may,  nf  necessity,  have  arise7i,  by  adopt- 
ing this  train  of  natural  points,  to  regulate  the  course  of  the 
preceding  part. 

Art.  7.  AnJ  this  consideration  is  equally  important  and 
decisive,  at  either  extremity  of  the  snid  l)otindary,  or  at  both; 
the  term  ^' final,"'  in  the  previous  article,  being  merely  adopted 
for  facility  of  expression  :  since,  of  the  two  extremities  of  any 
such  boundary,  either  may  be  taken  as  the  initial,  and  the 
other  as  the  fuial  extremity. 

Art.  8.  The  import  of  Articles  5,  6,  and  7,  conncctively 
taken,  is  expressible  thus.  In  the  prescription  of  any  such 
boundary,  the  total  course  of  the  proceedings  will  be  govern- 
ed by  this  consideration  :  so  to  accommodate  the  sevo'al  parts 
of  that  boundary  to  each  other,  as  to  give  to  the  total  boundary, 
precisely,  or  as  nearly  as  possible,  a  self-adjusting  cJiaracter  ; 
that  is,  that  all  infi'ingements  of  the  general  pre-undcrstood 
conditions,  which  are  unavoidably  wade  in  certain  parts  of 
that  boundary,  in  favor  of  one  party,  are  counteracted  and 
recompensed  by  other  infringements  which  are  elsewhere  made 
in  the  course  of  that  boundary,  in  favor  of  the  other  party. 

Art.  9.  Now,  as  we  have  observed  in  Article  1,  that  all 
boundaries,  if  not  intercepted  by  obstacles,  would  be  rectilin- 
eal ;  and  as  all  desirable  conditions,  concerning  the  relative 
situations  and  magnitudes  of  two  shares,  can  be  satisfied  by 
merely  accommodating  the  situation  and  direction  of  a  rec- 
tilineal boundary  to  each  particular  case:  it  follows,  that, 
whenever  the  separation  and  distinction  of  two  several  shares 
is  required,  a  straight  line  miglit  be  drawn  somewhere,  which 
would  satisfy  the  general  pre-understood  conditions  between 
the  parties.     Such  a  straight  line  we  shall  call  a  Noiinal. 

Art.  10.  From  Arts.  8  and  9  we  learn,  that  if  any  Iiounda- 
ry  irhaterer  between  two  shares  be  completely  self-adjusted, 
the  following  remarkable  relation  must  be  verified  between  it, 
and  every  normal  whatever,  appertaining  to  the  same  case 
between  the  same  two  }>arties.  In  Fig.  I,  let  N  n  be  a  nor- 
mal, and  ABCDEFG  be  a  self-adjusted  boundary,  Q  q  and 
R  r  being   exterior  boundaries  of  the  total  tract  which  com- 


i 


if^nr 


r 


prehends  Iho  two   shares;    then,  the  sum  of  all  the   areas 
!  f.  y.  >.  &c.,  which  me  intereepted  by  ;V  „,  and  those  part, 
otlhe  tortnous  boundary  which  Ho  on  o«»  side  of  >t  .s  .^.«i 
the  sum  of  «,  K  0.  ,,,  .to.,  which  are  intorcepted  hy  V  . , 
and  by  those  parts  of  the  tortuous  line  winch  he  on  the  other 
side  of  it      For,  if  the  two  sums  ho;,ncqu«l,  the  slra.ght  and 
the  tortuous  boundary  cannot  botU  satisfy  the  understood  con- 
ditions of  the  relative  magnitudes  of  the  two  shares ;  smcc 
by  the   adoption  of  one  boundary,   o„c  party  ^^^' ;/'''[ 
share  and    he  «(/,«■  a  fc.s,  than  by  the  adoption  of  the  „//.  r 
bon  iary  ;  hut  both  boundaries  do  satisfy  the  aforesaid  eondi- 
tbn  n^«,by  i.»  character  as  a  „on„„i,  and  the  tort.tons 
boutrdaryby  le,f-<ulj„.t.nc,a :  therefore  the  two  snn.s  nrnst 

AuT   11.  Whenever  a  straight  lino  bears  to  any  other  Ime 
such  a  relation,  with  respect  to  the  hUercepted  areas  as  tha 
which    N  n  bears  to  the  tortnous  boundary  ni  Art.  10,  we 
shall  call  that  straight  line,  a  cocqaator  to  tlie  other  hne 

Art    12.  But  yet,  anion-   tortuous  lines,  the  general  bear- 
ing or  progressive  tendency  of  one,  may  depart  from  that  of 
its  coec^iator,  much   more  than   that  of  another  tortuous  hne 
tm  il  coecnau,r.     Thus,  in  F.gs.  2  and  3,  the  general  be    - 
..gs  of  the  tortuous  lines  ABCBl^.  and  «     c  '/  c/  m..  , 
JnaJc  in  direction  with  their  coequators  ^^  ?^  Y' /T  I   I 
the  general  bearing  oi  ah  c  dcfg  h,  nr  F.g.  4,  js  rtscl    the 
tortuous  line  a  d  i,  intersecting  the  coequator  a  Jc,   ^/i  and 
CD  being  the  exterior  boundaries  comprelicndmg   the  total 
tract  which  contains  the  two  shares.     Whenever  the  coequa- 
tor  and  the  general  bearing  exactly  coincide,  the  two  extremi- 
ties of  the  coequator  and  those  of  the  tortuous  l^.o  wilU  .o 
exactly  coincide,  as  evidently  appears  by  Figs,  i  and  o      ilns 
relation  we  shall  express  by  saying,  that,  in  every  such  case, 
the  tortuous  line  preserves  perfect  affinity/  with  its  coequator ; 
but  in  other  cases,  the  want  of  such  atiinity  shall  be  express- 
ed  by  stating,  that  it  more  or  less  diverges  from  that  coequator. 
Art   13.  Before  proceeding  further,   it  is  well  to  consider, 
-   in  what  circumstance  this  greater  or  less  divergency   consists. 
Two  straight  lines,  which  coincide  in  direction,  make  no  an- 


r 


\:ml 


13 


^:4 


gle.  Now  in  Figs.  2  and  3,  the  coeqimtor  AF,  in  the  former 
case,  and  a  J',  in  the  latter,  coincides,  or  is  identical  with,  a 
straight  line  u)iiting  the  extremities  of  the  houndari/  to  which  it 
is  dratvn  as  a  cotj/natur.  In  oacli  cnso,  the  coL'([n;it(n'  makes  no 
angle  with  the  coiincctiiig  hue,  and  therefore  we  say  that  sncii 
boundary  has  no  divergence  iVotn  that  coe([iiator,  or  has  perfect 
ajjinitii  with  it,  by  which  we  mean  the  same  thing.  But,  in 
Fig.  4.  tiie  straight  hne  a  h,  wiiich  connects  the  extremities 
of  the  dotted  boundary,  does  make  an  analo  with  the  cociiua- 
tor  a  k,  and  therefore  Ave  say  that  such  boundary  divcnres 
from  tliat  coequator.  Lastly,  in  Fig.  5,  if  a  0  c  d  be  a  bounda- 
ry, and  a  d,  c  f  and  :,-■  /*,  be  all  coeqtiators,  AB  and  CD  being 
the  exterior  boundaries,  comprehending  the  total  tract  which 
contains  the  two  isiiares;  we  ])erceive  that  a  b  c  d  \\ns  no  di- 
vergence from  a  d,  but  it  has  a  divergence  from  ef  as  we  per- 
ceive by  the  angles  <-■  and  .-(.  which  cf  makes  with  a  d.  But 
ah  c  d  has  a  yet  greater  divergence  from  g  h ;  for  g  h  makes 
angles  with  a  d,  which  severally  exceed  the  two  former,  by 
the  angles  /  atid  a. 

AuT.  1-1.  But  here  an  indispensable  remark  must  be  made. 
For,  since  any  two  straight  lines,  which  meet,  will  make,  if 
produced,  as  in  Fig.  G,  foiir  angles  equal  by  j)airs,  as  a  and  ",  h 
and  i"*-  and  since,  if  either  pair  be  very  small,  the  divergence  of 
the  two  lines  will  be  very  small;  therefore,  of  two  adjacent 
angles,  which  one  straight  line  makes  with  another,  as  the 
two  angles  a  and  b,  in  Fig  7;  if  these  two  be  unequal,  the 
minor  angle  is  that  whereby  we  measure  the  divergence. 

Art.  15.  Now  as  it  appears  from  Arts.  1  and  9,  that  all 
boundaries,  if  not  eiicountered  by  natural  impediments,  would 
be  Normals,  because  boundaries  of  such  a  character  ofier  to  the 
human  understanding  the,  only  obvious,  primary,  and  imme- 
diate resource  whereby  to  realize  the  pre-conceived  conditions 
between  the  parties;  it  ''oUows,  that  where  a  boundary  has 
been  eventually  adopted  between  two  such  parties,  the  first 
notion  of  such  boundary  must  have  been  that  of  a  normal, 
which,  for  the  sake  of  acconnnodation  to  existing  circum- 
stances, has  been  diversified  and  altered  into  that  form  under 
which  it  was  finally  adopted.     In  every  such  case,  the  pre- 


u 

conceived  original  normal  we  shall  call  the  Archeii/pe.  and  the 
actnal  bonndary  therefrom  derived  M^e  shall  call  the  McUttype. 
_  Art.  1(3.   Hence  we  obtain  a  tost  for  dhe' ddcvtioa  of  a  spu- 
nous  houndcmj.     For  if  we  can  .colleet  any,,evidence  to  show 
that  such  boundary  has  been  obtained  and  derived  from  iw 
normal  archetype,  we  have  established  a  proof,  that  such  boun- 
dary was  never  adopted  as  the  result  of  the  mutual  delibera- 
tion of  two  contracting  parties,  but  betrays  a  fictitious  origin 
by  the  devious  a.-d  distorted  asj)ect  of  its  general  career;  pre- 
senting,  indeed,  no   unsuitable    picture  of  the   inconsistent, 
perverse,  and  guilty  movements  of  the  creating  spirit  ichich 
directed  it ;  alternately  impelled  and   restricted  by  tin;  stimu- 
lations of  encroaching  avarice,  and  the  retiring  trepidation  of 
abashed  and  conscious  turpitude. 

AuT.  17.  To  all  that  feel  disposed,  without  reflection,  to 
denounce  the  principle    detailed  in    the  prrcediny  Article,  as 
far-fetched  and  fanciful,  it  may  be  well  to  obscirve,  that  2/"  that 
principle  be  a  mere  metaphysical  and  idle  conceit,  the  fact  is 
very  extraordinary,  that,  from  the  most  primitive  and  imspec- 
ulating  ages  of  the  human  race,  down  to  the  present  time,  the 
universal  siracturc  of  lauLfuriL;-e  will  clearly  prove,  that,  from 
the  geometrical  ideas  of  "  SlraiL;iit,^'  and  ''Crooked,''  ''Even,'' 
and  "  Uneven,''  the  moral  ideas  of  ''Good,"  and  "£'r?7,"  have, 
in  virtue  of  the  right  of  kindred,  borrowed  their  verbal  habil- 
iments.    With  moderate  research,  and  possibly  with  some  sur- 
prise, those  objectors  will  discover,  that  "Equity,''  is  "Flat- 
ness," and  moral  Rcctitudeis.  moral  Straightncss.     They  will 
also  perceive  that  a  "  Delinrpumt"  is  one  who  swerves  from 
the  path  of  duty,  so  as  to  "  Leave"  that  path  lying  "Of"  or 
aioay  from  his  present  course.  They  will  also  further  perceive, 
that  a  "Perverse"  disposition  is  that  which  is  "Very  Much 
Tivisted  Away."     To  multi])ly  examples  of  this  kind  is  not 
necessary  ;  they  are  familiar  to  every  linguist,  and  clearly  in- 
dicate an  intrinsic  similarity  and  a  usual  association  of  the^e 
geometrical  and  moral  ideas,  for  which  no  satisfactory  reason 
can  be  easily  given,  if  we  deny  the  proposition,  that  a  tendency 
to   straightness    ivill   be   a  prevailing  feature  in  a  boundary 
traced  for  the  first  time  between  two  tracts  of  land,  under  the 
direction  of  honest  men. 


[H 


io 


AuT.  IS.  The  process  of  deriving  a  boundary  from  its  nor- 
mal archetype  will  be  evidently  guided  and  ruled  by  this  con- 
sideration ;  lo  shun  iiiis-reL,nd(iti(>ii,  perplexity,  and  toil,  hij  di- 
7'ccting'  this  boundari/,  so  as,  first,  to  render  it  self-adjusted, 
wJicrcbij  the  normal  arrlietijpe  loill  be  its  coe(/uutor :  and  sc- 
condlij,  so  as,  if  possible,  to  have  perfect  aflnitij  loitliits  nor- 
mal archetijpe,  or  else  to  diverge  front  that  archeti/pe  in  the 
S7nallest  possible  degree.  A  iluuiliar  illustration  of  this  case 
may  be  drawn  from  that  of  a  traveller,  who,  being  led  by  an- 
other person  through  an  unknown  country,  ivill  deviate  from 
the  trade  of  his  guide  as  little  as  possible. 

Art.  19.  Hence  we  obtain  a  satisfactory  test  whereby  we 
may  discover,  at  least  in  extreme  cases,  ichcther  any  boundary 
has  been  derived  from  a  normal  archetype,  or  fabricated  on  some 
other  principles.  For  if  we  can  discover  that  such  boundary 
diverges  extravagantly  from  each  of  all  its  possible  coequators, 
we  have  clearly  a  strong  presumption  that  such  boundary  was 
not  obtained  from  any  normal  archetype  ;  and,  on  the  other  hand, 
if  we  discover  that  such  boundary  has  07ic  coc(iuator  to  which 
it  has  perfect  affinity,  or  from  \vhich  it  very  slightly  diverges,  we 
have  an  ccfually  valid  presumption  of  the  opposite  kind. 

Art,  20.  But  a  question  may  occur,  of  the  same  general  na- 
ture, but  under  a  different  form,  wherein  the  preceding  test 
may  be  safely  applied,  not  only  in  extreme  cases,  but  in  any 
case.  For  if  two  boundaries  be  placed,  from  the  peculiar 
cause,  which  gave  them  origin,  or  from  any  other  circum- 
stance, under  such  conflicting  conditions  and  relations.,  one  to 
the  other,  that  one  of  these  boundaries  must  have  been  derived 
from  a  normal  archetype,  and  the  other  mmi  have  been  fabricated 
on  other  principles ;  luid  if  we  ascertain  that  we  can  draw 
to  one  of  these  boundaries  a  coequator  to  which  it  has  perfect 
affinity,  whereas  none  such  can  be  drawn  to  the  other;  or  else, 
if  we  draw  to  each  boundary  that  coequator  from  which  it 
has  least  divergence  ;  and  then  discover  the  divergence  of  the 
former  boundary  from  such  coequator  to  be  less  than  that  of 
the  latter  from  its  own  coequator ;  we  obtain  thus,  for  the  for- 
mer boundary,  a  cause  of  preference  before  the  latter,  which  de- 
cides the  question. 


16 


Art.  21.  It  is  also  obscrvablo    that 


sometimes  a  coeqiiator 
I  comircts  the  cx- 


may  be  iiaralld  to  tlio   straight  lino  whici 
ticinitios  of  the  boundary,  as  dc  to  «f,  wliid 

tremities  of  the  boundary  ahr,  in  Fig.  8:  or,  thou-d 

all  :.       ,      ,  - 


1  connects  the  ex- 


I  not  par- 


may  not  meet  tliat  straight  lino,  ,ml,ss  both  be  produced 


beyond  the  total  tract,  as  the  cooqnatOi  An-,  in  Fiir.  9   moots 


1"  4r, 


the 


exterior  borders   of  the  tract    b 


a  c 


AliCDE  and 


tract  uonig  ./imur^  am. 
All  such  coequators  we  shall  desirrnato,  for  the  sake 
of  dkstmction,  as  Remote,  and  all  others  as  Adjacent.  From  the 
extreme  dilliculty,  however,  of  directing  an  adjusted  boundary 
by  any  such  remote  coe.|uator,  it  is  needless  to  say  that  no 
boundary  has  ever  been  derived  from  such  a  cocquator  as  its 
normal  archetype.  In  the  examination,  therefore,  of  the  com- 
parative claims  of  dilFerent  boundaries,  to  a  genuine  origin 
from  a  normal  archetyjio,  we  iiave  no  cause  to  make  any  in- 
qmry  with  respect  to  remote  amjuators. 

Art.  22.  If  two  boundaries  be  so  circumstanced,  that  07ie 
must  be  spurious  and  the  other  ,irc7imne,  hnt— which  to  ac- 
knowledge as  genuine,— md  which  to   reject— wo  arc  uncer- 
tain ;  the  only  necessary  aid,  which  has  not  yet  been  furnished 
for  the  settlement  of  such  a  question,  is  a  process  whereby  we 
can  try  the  question  of  greater  or  less  divergency,  as  stated  in 
the  preceding  Articles.     For,  by  such  a  process,  we  can  pre- 
sumptively  ascertain,   as  appears  by  those  Articles,  tohich  of 
those  boundaries  has  been  derived  from  a  normal  archetype, 
and  which   was  fabricated  on  other  principles ;  and  again  by 
Art.  10,  from  the  discovery  of  these  latter  facts,  we  are  imme- 
diately enabled  to   discriminate    the    spurious  from   the  gen- 
nine  boundary. 

Art.  23.  From  each  of  the  two  extremities  of  any  bounda- 
ry a  coequator  to  that  boundary  may  be  drawn.  For  let  cmy 
straight  line,  as  de  in  Fig.  10  and  11,  be  drawn  from'one  ex- 
tremity, as  d,  of  a  boundary  abed.,  till  it  meets  the  opposite 
exterior  border  of  the  tract  in  .;  de  either  not  a^ain  meeting 
the  boundary,  as  in  Fig.  10,  or  again  meeting  it  one  or  more 
times,  as  in  Fig.  11.  In  the  former  case,  let  the  area  of  the 
space  comprehended  between  the  boundary  «icrf,  the  strai^^ht 
hne  dc,  and  the  border  ae,  be  calculated  and  represented^by 


A  ;  it  is  now  an  easy  goodosic  prol)Icm,  lo  dotcrminc  the  posi- 
tion of  a  straight  lino,  as  the  dotted  line  dn,  which  will  make 
the  space  comprehended  between  itself,  dc,  and  the  border, 
cqnal  to  A.  The  straight  line  d  a  is  the  re(|nircd  coe{|uator. 
For  since  d  c  nugmcnts  one  of  the  shares,  as  determined  by  the 
boniidary,  by  the  cpianlity  A:  and  again  da  diminishes  by  tho 
same  (jiuuitity  A,  tho  share  which  has  thus  been  augmented  ; 
that  share,  and  consc(|ucntly  the  opposite  one  also,  is  now  re 
stored  to  its  foruter  value  ;  that  is,  d  a  \^  a  cnoquator.  In  the 
latter  case,  let  all  the  areas  intercepted  l)y  d  c,  by  the  bounda- 
ry, and  by  the  opposite  border,  on  one  side  of  d  e,  be  collected 
into  one  sum  ;  and  let  all  such  areas  on  the  ofhers'iCiQofdc  be 
collected  and  summed  in  the  same  manner;  and  if,  on  one 
side,  no  iuleroepted  area  be  found,  the  sum  of  such  areas  on 
that  side  may  be  slated  as  e(/nal  to  zero  ;  take  the  dillerence  of 
Uic  two  smns,  and  denote  it  by  '\;  then,  upon  that  side  of  d e, 
A'herc  tlio  sum  of  the  iiUercepted  areas  is  greater  than  the 
other,  dmw  d f,  so  as  to  make  the  space  comprehended  be- 
tween itself,  dc,  and  the  liorder,  equal  to  -^r  then  for  the  same 
reasons  whicli  v/cre  assigned  in  the  preceding  case,  df  must 
be  a  coequator.  Every  cocquator.  which  meets  one  of  the  ex- 
tremities of  a  boundary,  shall  hereafter,  for  the  sake  of  distinc- 
tion, lie  styled  Conterminal,  and  every  other  cocquator,  Dis- 
tcrminal. 

Art.  21.  All  cnequators  to  the  same  boundary  must  meet 
and  intersect  within  the  exterior  borders  of  the  total  tract 
which  contains  the  two  h,hares.  For  if  two  straight  lines  meet 
on  such  exterior  border,  as  a  e  and  b  c,  Fig.  12,  which  meet  on 
the  border  CD  ;  or  if  they  mvctncithir  on  nor  within  such  bor- 
der, as  a  c  and  h  d,  Fig.  13  ;  then  one  of  the  two  shares,  deter- 
mined by  one  of  these  straight  lines,  is  augmented  by  the  other 
straight  line,  without  any  counteracting  diminution ;  therefore 
they  cannot  be  both  coe(|uators.  Therefore  any  two  coequa- 
tors  must  intersect  within  the  '    -ders,  as  a  h  and  c  d,  Fi";.  14. 

Art.  2;~.  And  hence  it  immemately  results,  that,  from  either 
extremity   of  a   boundary,    we   can   draw  hut   one  cocquator; 

3 


i 


that  IS,  lu   iiViity  liouiidary,   we  cmii  dmw    two,   and  only  tiU 
conttniiinal  a,cf/iui(ors  ;  and,  in  thu  cimii  of  pcr/cd  aj/initi/,  these 
two  anuciitc  in  direction,  and  lliiis,  in  a  certain  sense,  become, 
one. 

AiiT.  20.  Of  all   cocquators  drawn  to  the  same  boundary, 
remote  ones  excepted,  one  of  the  two  conterminal  coe(iUiitors 
is  that  wiiich  has  least  divergence.     For,  in  Fig.  15,  let  a  and 
l>  be  the  two  extremities  of  a  boimdary,  which  it  is  not  neces- 
sary here  to  reprv;sein  ;  let  a  c  and  b  d  be  the  two  conterminal 
eoequators,  intersecting  in  g -,  let  c  Tc  be  any  other  coequator 
whicii  is  not  remote,  intcrsccLing  rt  c  in  h,  and  b  d  in  i ;  be- 
cause c  /.;  is  not  remote,  it  meets   the  connecting  line   a  b,  at 
some  pointy;  between  a  and  b;  now  of  the  triangle />/ 1,  the 
exterior  angle  ^  is  greater  than  the  angle  at  b;  and  of  the  tri- 
angle (if  h  the  exterior  angle  ^t  iy  greater  than  the  angle  at  a; 
therefore,   of  ,i  and  ,^,   the  7nino/'  one,  if  these  be  unequal,  or 
each  of  them,  if  cfiual,  is  greater  than  one  of  the  two  interior 
angles  at  a  and  b;  that  is,  in  any  case;  one  of  the  two  interior 
angles  at  a  and  b,  is  kss  than  each  of  the  angles  at  «  and  /<. 
<.)f  that  interior  angle  and  its  adjacent  exterior,  if  the  interior 
be  the  minor  angle,  or  if  both  be  equal,  the  conterminal  coe- 
quator, to  which  that  angle  appertains,  must  have  less  diver- 
gence than  c  Ic ;  but  if  the  adjacent  exterior  angle  be  the  mi- 
nor, then,  since  the  interior  angle  is  less  than  each  of  the  two 
angles  «  and  ,-;,  a  fortiori,  the   exterior  angle,  which  is  noio  the 
minor,  is  less  than  each   of  the  angles  .^  and  ,'<;  therefore,  in 
every  possible  case,  one  of  the  two  conterminal  coequators has 
less  divergence  than  e  k.     Q.  E.  D. 

AuT.  27.  Hence,  if  the  contlicting  claims  of  two  boundaries 
be  such,  that  one  must  be  spurious  and  the  other  genuine,  we 
have  the  following  practical  rule  for  a  decision  of  the  case. 
To  one  boundary,  draw  both  its  conterminal  cocquators,  and  ascer- 
tain the  divergence  of  each,  selecting  the  lesser  divergence,  or  ei- 
ther, if  equal,  as  the  least  possible  ivhich  that  boundary  can  have 
with  any  coequator  ivhich  is  not  remote ;  in  like  manner,  discover 
the  least  divergence,  in  the  case  of  the  other  boundary;  compare 
these  two  results  together ;  then,  whichever  bound  try  affords  the 


lU 


less  result,  wc   obtain,  by  tlic.  principles  ilctailcd  in  the  prccnlinL; 
articles,  a  in'riiondcrating  evidence  in  favor  of  that  buiindari/. 

Art.  28.  Uy  tlio  mere  inoiitiil  siibstitiitioii  of  arcs  of  %rcat 
circles  for  straii^ht  lines,  the  whole  preceding  theory,  with 
scarcely  a  verbid  alteration,  becomes  tlircctly  appHcibb!  to  all 
cases,  in  wliich,  from  the  large  extent  of  the  total  tract,  wo 
may  suppose  the  rotundity  of  the  terrestrial  surface  to  have 
any  perceptible  cll'ect  on  the  (jncstion. 

Art.  21).  But,  in  the  practical  ap[)lication  of  t!iis  theory, 
the  following  indispensable  ])rccaMtion  must  be  observed. 
Through  all  the  jnx'coding  investigations,  the  extent  of  any 
one  boundary  is  conceived  to  be  that  wiicrcin  it  separates, 
every  loherc,  the  preciovsly  undetermined  shares  of  those  two  par- 
ties, and  of  those  two  parties  alone.  And  thercibrc,  if  a  bounda- 
ry be  part  of  a  continuous  frontier  ;  and  if  that  frontier,  in 
other  parts  of  its  course,  bo  a  boundary  to  shares  predetermin- 
ed between  the  same  two  parties,  or  a  boundary  at  all  apper- 
taining to  any  share  of  any  third  party  ;  if  wo  would  apply 
the  preceding  investigation  to  this  first  boundary,  we  must 
first  ascertain  how  much  of  that  frontier  constitutes  this  boun- 
dary, by  discovering  on  what  points  of  that  frontier  the  cxtrcin- 
ilies  of  this  hoxindanj  fall. 

Art.  20.  Hence,  in  the  application  of  the  preceding  process 
to  an  examination  of  the  two  Imundarios  respectively  claimed 
by  the  American  and  British  nations,  on  the  frontiers  of  New 
Brunswick  and  Maine  ;  we  should  first  ascertain  vhcre  the 
common  westward  extremitij  of  the  two  conflicting  hounddvicfi 
lies :  that  is,  to  what  c.vtenl  westward,  in  178'J,  the  bonndnr;/ 
established  in  that  year  separated  British  from  American  ter- 
ritory. The  true  answer  to  this  (piestion  seems  to  be,  that 
such  boundary  was  honafule  a  boundary  between  the  United 
States  and  the  British  possessions,  through  all  its  westward 
course  till  it  first  reached  a  branchof  the  Mississippi:  as  every 
part  of  the  region  westward  of  that  river,  at  least  as  flir  as  lo 
the  Rocky  Mountains,  appears  to  have  been,  at  that  time,  ei- 
ther Indian  or  French  territory.  This  question,  however,  be- 
ing subject  to  the  decision  of  historical  research,  ils  I'lutlier 
consideration  is  unncccssarv  here. 


20 


AN  EXAI\TI\yVTION  OF  THE  TRDATY  Oi- 


1783 


ruoti  WHICH 


IS    DKniVED 


A  DEMOXSTRATION  OF  THE  FALSEHOOD  OF 
THE    ]3RlTISn   CLAIM, 


AND    ALSO    or 


THE  TRUTH  OF  THE  AMERICAN 


Am.  I.  An  a„n-/c  of  any  siipcificial  fi-urc,  lies  at  a  i)oint 
where  two  sides  of  the  figure  meet,  as  the  point  A  in  Fig.  10. 

Art.  2.  Two  sides  of  any  such  iigurc  can  meet  onhj  at  their 
extremities. 

Art.  3.  Hence  the  angles  of  any  such  figuio  can  exist  only 
at  the  extremities  of  the  several  sides. 

AuT.  .1.  llouco,  any  sucli    ligure  has  no  angle  at  a  point  on 
any  of  its  own  sides,  hclwecn  the  extrctnities  of  that  side. 

Art.  5.  it  may  easily,  liowevcr,  have  an  angK;  o{  nn^iher 
figure,  or  several  anyles  of  as  many  several  fmurcs,  :'t  ^uch  a 
point  on  its  own  sidi.",  between  the  extremities  of  that  side;  as 
the  angles  a  and  r,  :-  the  point  h,  in  Fig.  17,  vrliere  AB,  BC, 
CD,  and  DE,  aio  si.^  !S  of  a  figure  wluu-eof  the  total  repre- 
sentation is  not  necessary  here;  and  h  d  is  a  common  side  of 
two  other  figures,  whose  representation  is  also  unnecessary. 
.  Art.  G.  Before,  during,  and  after  tlio  preparation  of  the 
treaty  of  17S3,  for  a  certain  space  of  time,  the  two  tracts  of 
country,  which  are  now  called  Nova  Scotia  and  New  Rriins- 
wick,  were  comprehended  under  the  common  appellation  of 


21 

Nova  Scotia.      Sec  the  article,  '^  Nova  Scotia,''  in  the  En- 
cyclopedia Americana,  and  also  in  Dr.  liccs's  Cyclopedia. 

Art.  7.  In  that  comprehensive  sense,  we  shall  employ  the 
term  in  this  examination. 

Art.  8.  Now  it  np[)cars  from  the  map  of  Lower  Canada, 
with  adjacent  parts  of  the  United  States  and  Nova  Scotia,  pnb- 
lishcd  hy  the  Jlrithh  "Society  for  the  Dilhision  of  Usefnl 
Knowledi'e,''  that  one  side  of  Nova  Scotia,  namely,  the  iccst- 
ern,  is  a  line   extending  Aurlh  and  Soutfi,  as  BC  in  Fiir.  17  : 


wh 


hose  northern  extrennty  C  meets  another  sidt;,  namely,  CD, 
and  its  sonthern  extremity  meets  the  Chiputnaticook  river, 
which,  by  the  line  of  its  progress  to  Passamacpioddy  Bay, 
makes  a  t/iird  side.  It  also  appears  from  that  map,  that,  ac- 
cording to  the  British  claim,  the  "  nortlnvesl  uniilc  of  Nova 
Scotia;'  specified  in  the  treaty  of  1783,  lies  on  the  point  b, 
of  the  side  BC,  between  the  extremities  of  that  side,  and  b  d 
is  a  common  side  or  boundary  of  t)ic  United  States  and  of 
Canada.  But,  by  what  wc  have  shown  in  articles  4  and  5, 
the  angle  a  would,  in  that  case,  be  the  southeast  angle  of  Can- 
ada, and  the  angle  c  Avould  be  the  northeast  angle  of  the  Uni- 
ted States:  but  Nova  Scotia  has  no  angle  whatever  at  the 
point  b;  therefore  the  British  claim  must  bt  false. 

Aht.  9.  But,  by  the  American  claim,  the  northwest  angle 
of  Nova  Scotia  lies  at  the  point  C,  where  undoubtedly  there 
is  an  angle  of  Nova  Scotia ;  which  angle,  being  also  the  only 
angle  of  Nova  Scotia  which  is  made  by  the  northern  extremi- 
ty of  the  side  JW;  it  must  be  the  northwest  angle  of  Nova 
Scotia,  specified  in  the  treaty  aforesaid.  Therefore,  the  Brit- 
ish claim  is  false,  and  also  the  American  is  true.  Q. 
E.  D. 


I  cannot  conclude  the  foregoing  investigations,  without  ac- 
knowledging my  obligations  to  Dr.  J.  1).  Hedge,  of  Cam- 
bridge, for  the  kind  and  active  manner  in  which  he  has  labored 
to  ensure  the  correctness,  encourage  the  publication,  and  pro- 
mote the  success  of  this  work,  by  the  removal  of  many  prac- 


22 

tical  difficulties,  which,  otherwise,  if  not  insurmountable, 
would  have  created  extreme  delay  and  embarrassment  ;  mor(3 
especially  I  refer  to  the  exertions  of  that  gentleman  for  the 
facilitation  of  my  access  to  authorities  and  documents,  which 
I  should  otherwise  have  found  it  extremely  difficult,  if  not 
impossible,  to  procure. 

Finally,  Sir,  having  now  completed  a  task,  in  the  perform- 
ance of  which  I  have  been  animated  only  by  an  impulse  of 
duty,  combined  with  an  encouraging  hope  of  inheriting  a  share 
of  that  immortal  celebrity,  which  the  man  who  benefits  his 
race  by  intellectual  achievements  may  extort  from  the  grasp 
of  an  unwilling  world,  and  retain  to  the  latest  extent  of  hu- 
man posterity  ;  • 

I  subscribe  myself, 

with  great  respect, 

your  Excellency's 
obedient  servant, 

JOHN  LEE. 


\ 


\ 
',    i 


